Series Expansions
Contents
- Maclaurin Series
- Other Series
Maclaurin Series
Throughout this subsection
ζ
(
k
)
is
1
Γ
(
z
)
=
∑
k
=
1
∞
c
k
z
k
where
c
1
=
1
,
c
2
=
γ
, and
(
k
-
1
)
c
k
=
γ
c
k
-
1
-
ζ
(
2
)
c
k
-
2
+
ζ
(
3
)
c
k
-
3
-
…
+
(
-
1
)
k
ζ
(
k
-
1
)
c
1
k
≥
3
.
For 15D numerical values of
c
k
see
Abramowitz and Stegun(1964)(p. 256), and
for 31D values see
Wrench(1968).
ln
Γ
(
1
+
z
)
=
-
ln
(
1
+
z
)
+
z
(
1
-
γ
)
+
∑
k
=
2
∞
(
-
1
)
k
(
ζ
(
k
)
-
1
)
z
k
k
|
z
|
<
2
.
ψ
(
1
+
z
)
=
-
γ
+
∑
k
=
2
∞
(
-
1
)
k
ζ
(
k
)
z
k
-
1
|
z
|
<
1
,
ψ
(
1
+
z
)
=
1
2
z
-
π
2
cot
(
π
z
)
+
1
z
2
-
1
+
1
-
γ
-
∑
k
=
1
∞
(
ζ
(
2
k
+
1
)
-
1
)
z
2
k
|
z
|
<
2
,
z
≠
0
,
±
1
.
For 20D numerical values of the coefficients of the Maclaurin series for
Γ
(
z
+
3
)
see
Luke(1969)(p. 299).
When
z
≠
0
,
-
1
,
-
2
,
…
,
ψ
(
z
)
=
-
γ
-
1
z
+
∑
k
=
1
∞
z
k
(
k
+
z
)
=
-
γ
+
∑
k
=
0
∞
(
1
k
+
1
-
1
k
+
z
)
and
ψ
(
z
+
1
2
)
-
ψ
(
z
2
)
=
2
∑
k
=
0
∞
(
-
1
)
k
k
+
z
Also,
ℑ
ψ
(
ⅈ
y
+
1
)
=
∑
k
=
1
∞
y
k
2
+
y
2
